control of bistability, oscillations and chaos in a model

Хімія, фізика та технологія поверхні. 2011. Т. 2. № 3. С. 347–353 _____________________________________________________________________________________________

UDС 530.182

CONTROL OF BISTABILITY, OSCILLATIONS AND CHAOS IN A MODEL ELECTROCATALYTIC PROCESS ON A SPHERICAL ELECTRODE SURFACE O.I. Gichan Chuiko Institute of Surface Chemistry of National Academy of Sciences of Ukraine 17 General Naumov Street, Kyiv 03164, Ukraine The role of a preceding chemical reaction in the appearance of three dynamical instabilities leading to multistability of steady states, periodic current oscillations, and possible chaotic behavior in a model electrocatalytic process on a spherical electrode surface under potentiostatic conditions has been determined. A class of non-equilibrium systems where these instabilities can arise was sketched out and possible applications of these instabilities were presented from recent literature data. The results obtained are actual and fruitful in context of nonlinear dynamics and can help to understand the processes at interfaces. INTRODUCTION

playground for testing different control strategies. This not only helps to unravel the origin of instabilities in electrochemical systems but also can help to shed light on mechanisms of such selforganization phenomena in other non-equilibrium systems, including those of practical relevance [29–31]. In the paper, the method of impedance spectroscopy is applied to quantitatively characterize three dynamical instabilities, namely, a Hopf instability, a saddle-node instability, and a homoclinic instability in a model electrocatalytic process on a spherical electrode which is preceded by a homogeneous chemical reaction in bulk solution under potentiostatic control.

The first observation of spontaneous periodic oscillations during an electrochemical surface reaction is dated back to 1828. As earlier as 1900 electrochemical oscillations indicated chaos was reported, but at that time this term had not yet been used. Since then the experimental evidence of electrochemical instabilities leading to various types of nonlinear behavior including multistability, periodicity, quasiperiodicity, and chaos have been constantly accumulating [1, 2]. The mentioned nonlinear effects can be undesirable, but in some cases they can be exploited as to obtain new functional materials, including nanostructured ones [3], to increase reaction rates and yields [4–12], to devise an electrode with certain properties [13, 14]. The experimental and theoretical interest to these questions has also been renewed with the appearance of a new paradigm in chaos theory – the possibility to control chaotic behavior of a non-equilibrium system by application of small perturbations to it [15]. Today there is an attempt to use the patterns of chaos to encode and to manipulate inputs so as to produce a desired output. The resultant morphing logic gates are termed chaogates [16]. Chaos, including well-defined transitions to chaos, has been observed numerous times in electrochemistry [17–28]. As the dynamical instabilities in electrochemical systems can be monitored by simple current or voltage measurements and parametric changes can be easily achieved by current or voltage control, electrochemical systems seem to be an ideal

THEORETICAL PART The model process can be schematically written as k k K  → A bulk → A surf ← → A ads → Bbulk ← P + e, (1) a

kd

e

where k1, k2 are the rate constants of chemical reaction in the bulk solution, and ka, kd, Ke are the rate constants of adsorption, desorption, and electron transfer, respectively. The last step of the reaction is considered to be irreversible. This model electrocatalytic reaction was proposed for a theoretical explanation of spontaneous current/potential oscillations during electrooxidation of small organic molecules such as those of formaldehyde, methanol, ethanol, ethylene glycol, etc. on Pt electrode and other noble ones. These molecules were tested for their pos-

* corresponding author [email protected] ХФТП 2011. Т. 2. № 3

1

k2

347

O.I. Gichan _____________________________________________________________________________________________

∂c(r , t ) 1 ∂ ∂c(r , t ) = D 2 (r 2 ) − kc (5) ∂t ∂t r ∂r with boundary conditions ∂c(r , t ) J c (r0 , t ) = − D r =r0 = −ν1 (t ) , ∂r (6) c (δ , t ) = c0 , δ = r0 + d , where k is the effective rate of a preceding homogeneous chemical reaction, D is the diffusion coefficient, d is the Nernst diffusion layer thickness. The origin of coordinates coincides with the center of the spherical electrode. The faradaic current density is determined by the following equation:

sible application in fuel cells [32–34]. We added into the stage of a preceding chemical reaction in a near-electrode layer the reaction scheme [35–39]. Our interest was motivated by similar studies on influence of a preceding chemical reaction on impedance spectra [40]. Neglecting the Ohmic losses and double layer influence, we can write the rates of adsorptiondesorption and electron transfer as follows:

ν1 (θ (t ), c(r0 , t )) = Γka exp(γθ (t ) / 2) c(r0 , t ) × × (1 − θ (t )) − Γkd exp(−γθ (t ) / 2)θ (t ),

(2)

ν 2 (t ) = ΓK e (t )θ (t ) = Γk e exp(α bE (t ))θ (t ) . (3)

i f (t ) = Fν 2 (t ) = F Γke exp[α bE (t )]θ (t ) .

Here, c( r0 , t ) is the concentration of electroactive species at the electrode surface, r0 is the radius of the spherical electrode, θ(t) is the coverage of the electrode surface with the adsorbate, γ is the attraction constant, Γ is the maximum surface concentration at θ(t) = 1, F is the Faraday’s number, R is the gas constant, T is the absolute temperature, α is the symmetry factor of electron transfer in the direction of oxidation, E is the electrode potential, b = F/RT, ke is the rate constant of electron transfer at E(t) = 0. According to Eq. (2) the adsorptiondesorption of species A on the electrode surface under steady-state conditions is described by the Frumkin isotherm [41]. Positive values of the attraction constant γ correspond to the attraction between adsorbed species, and its negative values correspond to the repulsion between adsorbed species. Only positive value of the attraction constant leads to instability [32–34]. In general, γ depends on potential [42, 43]. The adsorption equilibrium constant ka/kd can also be a function of the potential. We considered a case when γ is a constant and the adsorption-desorption rate constants are independent on the potential. As the control parameters we take γ and the effective rate of a preceding homogeneous chemical reaction k. The changes in the electrode surface coverage θ with adsorbate and in the concentration c(r, t) = c0 + u (u is a deviation of concentration from equilibrium concentration c0 corresponding to bulk concentration) satisfy the following equations: dθ (4) Γ = ν1 (t ) − ν 2 (t ), dt

(7)

Steady-State Condition. The solution of the diffusion equation (5, 6) under steady-state conditions yields the following expressions for the steady-state electrode potential Est, the steadystate concentration at the electrode surface cst(r = r0) and the steady-state faradaic current density ifst m* (c − c (r )) Est = (α b)−1 ln[ c 0 st 0 ] , (8) Γkeθ st

cst (r0 ) =

mc*c0 + Γkd θ st e

−γθ st / 2

mc* + (1 − θ st )Γka e

γθ st / 2

i fst = Fmc* (c0 − cst (r0 )) , where G0 =

mc* = mc

th τ d k

τd k

1 + ε G0 , G0

, τd =

mc =

D , d

,

(9)

(10)

ε=

d , r0

d2 ( τ d is diffusion time of D

relaxation). As in the calculations we neglected the electrolyte resistance and did not take into account the double layer impedance, that of the model system is equal to the faradaic one. Hopf bifurcation and impedance spectra. It is known that the origin of electrochemical instabilities stems from the negative impedance characteristics of the faradaic processes at the electrode surface [32–34]. For potentiostatic conditions the study of the linear stability of an electrochemical system near its steady state is based on the analysis of the system impedance zeroes with changes of electrode potential [44]. The sign of the real part of impedance zero indicates the stability region of the stationary state [45]. As was 348

ХФТП 2011. Т. 2. № 3

Control of bistability, oscillations and chaos in a model electrocatalytic process on a spherical electrode surface _____________________________________________________________________________________________

∂θν1 = Γ{kd exp( −γθ st / 2)[γθ st / 2 − 1] +

shown, a Hopf bifurcation can realize when complex impedance of electrochemical system is equal to zero at nonzero frequency, Z(ω) = 0 at ω = ωH ≠ 0 (ω = 2 πf, f – frequency) [32–39, 44–49]. At Hopf bifurcation the system produces its own undammed periodic oscillations with frequency ωH, so in the case of influence on the system of an external signal with a frequency exactly coinciding with this value, the external signal will pass through the system without resistance. In many cases a Hopf bifurcation is a prerequisite for more complex dynamical behavior, including chaos. To calculate the faradaic impedance of this system, we consider its dynamic behavior when a low periodical signal is applied to the steady-state polarization potential

E (t ) = Est + ∆E0e jωt ,

(18) (19)

(20) For sake of convenience, the partial derivatives are designated here as ∂ x u = ∂u / ∂x . Thus, to find the Hopf bifurcation points of the system, the zeros of impedance Z f ( s ) depending on electrode potential must be studied. In our case, the impedance zeros can be found from the following equation: Ψ ( k , s,θ ) = (Γs + ∂θν 2 )[1 + ε G (1 + µ∂ cν1 )] − (21) −∂θν1 (1 + ε G ) = 0. This is the equality to zero of the numerator of Eq. (14). To satisfy the condition (22), it is necessary that  Re[Ψ ( s,θ , k )] = 0 (22)   Im[Ψ ( s,θ , k )] = 0 .

(11)

Eq. (22) can be solved only numerically. The intersection of the obtained surfaces gives the Hopf bifurcation points, namely, bifurcation values of frequency ωH, electrode surface coverage by adsorbate θH and the effective rate of a preceding chemical reaction kH. In order to pass from the Laplace space to the Fourier space, it is necessary to perform a substitution s = jω. Saddle-node bifurcation and polarization resistance. The saddle–node bifurcation indicates the realization in a non-equilibrium system bistability – the coexistence of two stable steady states at the same values of the parameters [32–34, 36, 49]. The bistability region is that where the system can be in one of two stable steady state. Which one is chosen depends on where the system comes from, i.e. on initial conditions. In this sense the bistable system has a memory. The saddle-node bifurcations always come in pairs and lead to hysteresis. The saddle–node bifurcation points can be found from equality to zero of an electrochemical system polarization resistance. As well known, polarization resistance is equal to the system impedance at ω → 0 , Z P = lim Z f (ω ) .

(12) c ( r0 , t ) = cst ( r0 ) + ∆c ( r0 ,θ ) . The faradaic impedance in the Laplace image ∞

space F ( s ) = ∫ f (t )e − st dt is expressed as a func0

tion of complex frequency: (13)

Omitting some mathematical computations, the resultant expression for the faradaic impedance in the Laplace space can be written as: Z f (s) = ∂θν 2 [1 + ε G(1 + µ∂cν1)] (14) = Rct {1 + }, Γs[1 + ε G(1 + µ∂cν1)] − ∂θν1(1 + ε G) where 1 1 (15) Rct = = ∂i fst F Γα fke exp(α fEst )θ st ∂Est (charge transfer resistance), th τ d (k + s ) G= (16) τ d (k + s ) (finite length Gerischer impedance),

ХФТП 2011. Т. 2. № 3

∂ cν1 = Γka (1 − θ st ) exp(γθ st / 2) ,

µ = r0 / D .

θ (t ) = θ st + ∆θ0e jωt , i f (t ) = i fst + ∆i f ( E ,θ )

∆E ( s) . ∆ i f (s)

(17)

∂θν 2 = Γke exp(α bEst ) ,

where j = −1 , ω – angular frequency (ω = 2 πf, f – frequency), ∆E0 – amplitude of small periodic signal. In response to this excitation the electrode surface coverage θ(t) oscillates near steady-state value as the faradaic current if(t) and concentration c(r0, t) do.

Z (s) =

+ka exp(γθst /2)cst ( r0 )[γ (1−θst )/2−1]},

ω →0

In our case the system polarization resistance has the form ∂ ν [1 + ε G1 (1 + µ∂ cν1 )] Z P = Rct {1 + θ 2 }, (23) −∂θν1 (1 + ε G1 ) 349

O.I. Gichan _____________________________________________________________________________________________

where G1 = th τ d k / τ d k . To find bifurcation values of electrode surface coverage by adsorbate θSN1 and θSN2 that confine the bistability region, we should solve Eq. 24 (24) Z P = 0. This can be done only numerically. Homoclinic bifurcation – complex dynamics. Since Poincaré's pioneering work, orbits which are homoclinic to saddle periodic ones have been one of the most attractive objects under study in the theory of dynamical systems. The main reason to study homoclinic orbits is that their presence implies complicated dynamics, including chaotic one [20, 34, 50–52]. A homoclinic bifurcation occurs if an oscillatory state collides with a stationary state of the saddle type (in phase plane of dynamical variables at critical parameter value the limit cycle – oscillatory solution of the system born in a Hopf bifurcation, connects to the saddle in a homoclinic orbit). The homoclinic orbit is an oscillatory state of infinite period. When approaching a critical parameter value, the period of the oscillations will grow logarithmically, beyond the bifurcation, oscillations no longer occur. In our case a homoclinic bifurcation is realized in the system when the bifurcation values of the electrode surface coverage by adsorbate θ SN1 and θ H approach each other. In the model calculations, the following values of the system parameters were taken: Γ = 10-9 mol·cm-2; γ = {5; 8}; Γka = 0.1 cm·s-1; Γkd = 10-5 mol/cm2·s; ke = 10 s-1; D = 10-5 cm2/s; d = 10 -3 cm; α = 0.5; c 0 = 10 -5 mol/cm3 ; F = 96484 C/mol; R = 8,314 J/mol·K; T = 300 K; b = 38,7 V-1, r0 = 10-2 cm. All numerical calculations were performed with the mathematical package Mathematica™ [53].

called negative differential resistance (NDR) region where a Hopf instability usually occurs. In the second case the distorted steady-state polarization curves have also a region of bistability, where two values of the faradaic current density corresponds to one value of the electrode potential. In both cases an increase in the preceding chemical reaction rate, k, leads to an increase in the faradaic current density as for a plane electrode [35–39]. Fig. 1 shows the surfaces of zeros of the functions Re[ Ψ ( s ,θ , k )] and Im[ Ψ ( s ,θ , k )] for the chosen values of the attraction constant γ. The line of their intersection is the solution of Eq. (22) – the Hopf bifurcation points of the system.

а

б Fig. 1. Surfaces of zeros of the system impedance real (light) and imaginary (dark) parts: a – γ = 8 , b – γ=5

As it is seen from Fig. 1, in case of the higher value of the attraction constant we can always find two Hopf bifurcation points at a fixed value of the parameter k , up to its threshold value kth . For the values of k higher than this threshold value kth (kth ≈ 132 s-1) the system remains stable to the Hopf bifurcation, and periodic oscillations are absent. For γ = 5 the threshold value kth (kth ≈ 16 s-1) is lower and the system has only one Hopf bifurcation point up to this value of preceding chemical reaction rate. An increase of k shifts the bifurcation frequency ωH to the region of lower values in contrast to the case of γ = 8.

RESULTS AND DISCUSSION We have an electrochemical system of the N-NDR type [2]. In this system the electrode potential plays the role of an activator and the concentration of electroactive species in the nearelectrode layer plays the role of an inhibitor. We considered changes in the system dynamics at two values of the attraction constant, namely, γ = 8 and γ = 5. In both cases the steadystate polarization curves ifst vs. Est of the model process have a region of the potentials with the negative slope, where di f / dE < 0 . This is so350

ХФТП 2011. Т. 2. № 3

Control of bistability, oscillations and chaos in a model electrocatalytic process on a spherical electrode surface _____________________________________________________________________________________________

CONCLUSION On the basis of the theory of electrochemical impedance spectroscopy the necessary conditions for the appearance of three dynamical instabilities causing multistability of steady-states, periodic and chaotic oscillations has been obtained in a model electrocatalytic process on a spherical electrode surface under potentiostatic conditions. It was shown that the emergence of some of these dynamical instabilities may be regulated by the rate of the preceding homogeneous chemical reaction in the bulk solution. The spontaneous periodic oscillations can arise in the system at the rates of this reaction which are lower than some threshold value, kth, whereas at higher values of this parameter the system does not undergo such an instability. The appearance of bistability in the system is determined by the value of the attraction constant, γ, in the Frumkin isotherm and is less dependent on the values of parameter k. At γ = 8 we can observe only spontaneous periodic oscillations, while at γ = 5, both oscillations and bistable states are possible in the system and even more complex dynamical behavior evoked by realization of a homoclinic bifurcation.

Fig. 2. The system polarization resistance Zp vs. θ at a given value of the parameter k

Figure 2 presents dependence of the system polarization resistance ZP on the value of θ at a given value of the parameter k. Intersection of the ZP(θ) curve with the abscess axis yields two bifurcation values of the electrode surface coverage by adsorbate at which a saddle – node bifurcation appears in the system (the system impedance turns to zero at the zero frequency). A change in parameter k effects practically no changes in the bifurcation values θSN (see Tables 1 and 2). At γ = 8 the saddle– node bifurcation is not realized in the system – the ZP(θ) curve does not cross the abscess axis, as the shape of the voltammetric curve changes, i.e. there is no bistability. Table 1. The electrochemical system parameters in the Hopf and the saddle–node bifurcation points at k = 0.5 s-1 and γ = 5 Points of bifurcation

Hopf saddlenode saddlenode

ω,

θ

Hz

28.72 0.6632

ifst, А сm-2

Est, V

0.002891

0.077930

0

0.6496

0.002836

0.078008

0

0.2904

0.000420

0.020918

REFERENCES 1.

2.

3.

Table 2. The electrochemical system parameters in the Hopf and the saddle–node bifurcation points at k = 15 s-1 and γ = 5 Points of bifurcation

Hopf saddlenode saddlenode

ω,

θ

Hz

10.61 0.6505

ifst, А сm-2

Est, V

0.003868

0.093974

0

0.6491

0.003860

0.093974

0

0.2903

0.000569

0.036654

4.

5.

6.

Tables 1 and 2 indicate that the bifurcation values of electrochemical parameters in the Hopf bifurcation point and one of the saddle–node bifurcation points are very close. Such a feature of the system is retained in the whole range of values of the parameter k < kth. This is the case of positive implementation of a homoclinic bifurcation into the system as for a plane electrode [36]. ХФТП 2011. Т. 2. № 3

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Контроль бістабільності, осциляцій та хаосу в модельному електрокаталітичному процесі на поверхні сферичного електроду О.І. Гічан Інститут хімії поверхні ім. О.О. Чуйка Національної академії наук України вул. Генерала Наумова 17, Київ 03164, Україна, [email protected] Визначена роль попередньої хімічної реакції у появі трьох динамічних нестійкостей, що ведуть до мультистабільності стаціонарних станів, періодичних осциляцій струму і можливої хаотичної поведінки в модельному електрокаталітичному процесі на поверхні сферичного електрода за потенціостатичних умов. Представлено короткий огляд класу нерівноважних систем, де виникають такі нестійкості та розглянуто їх можливе застосування з останніх літературних даних. Отримані результати є актуальними та корисними в контексті нелінійної динаміки і можуть допомогти у розумінні процесів на межі фаз.

Контроль бистабильности, осцилляций и хаоса в модельном электрокаталитическом процессе на поверхности сферического электрода О.И. Гичан Институт химии поверхности им. А.А. Чуйко Национальной академии наук Украины ул. Генерала Наумова 17, Киев 03164, Украина, [email protected] Определена роль предшествующей химической реакции в возникновении трех динамических неустойчивостей, которые ведут к мультистабильности стационарных состояний, периодическим осцилляциям тока и возможному хаотическому поведению в модельном электрокаталитическом процессе на поверхности сферического электрода при потенциостатических условиях. Представлен краткий обзор класса неравновесных систем, где возникают такие неустойчивости, и рассмотрено их возможное применение из последних литературных данных. Полученные результаты являются актуальными и полезными в контексте нелинейной динамики и могут помочь в понимании процессов на межфазной границе. ХФТП 2011. Т. 2. № 3

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